Introduction to Viscous hydrodynamics

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Viscous hydrodynamics for relativistic heavy-ion
collisions
Huichao Song
Department of Physics
The Ohio State University
191 West Woodruff Avenue
Columbus, OH 43210
2007 National Nuclear Summer School
FSU, FL. July 8 July 21, 2007
Advisor U. Heinz, Supported by DOE
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The QCD Phase Diagram
QGP
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The QGP Shear Viscosity
The QGP shear viscosity of is in hot theoretical
debates recently
-Weakly coupled QCD prediction
P.Arnold,G.Moore,Y.Yaffe, 00,03
-Strongly coupled AdS/CFT prediction
D.T. Son, et,al. 03
The first estimation of the QGP shear viscosity
from the experimental data (RHIC)
--blast wave model the first order theory
corrections
D. Teaney, PRC 68 (2003) 034913
Blast wave model analysis
-only counts the viscous corrections to the
spectra during freeze-out procedure
-neglects shear viscosity effects on hydro
dynamic evolution before freeze-out
Viscous hydrodynamics is needed!
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Ideal hydrodynamics

hydro
S.Bass
Hydrodynamics
-A macroscopic tool to describe the expansion of
QGP or hadronic matter
Conservation laws
5 equ. 14 independent variables
-reduce of independent variables (ideal hydro)
-Provide more equations? (viscous hydro)
Ideal hydrodynamics reduce the independent
variable from 14 to 6,
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Causal viscous hydrodynamics
Full tensor decomposition in frame of
Besides these modified transport equations for
, one need more equations
for these dissipative flows
The first order theory formulism
--- Eckart, Landau Lifishitz around 1950s
Problems in the first order theory formulism
-Acuasl, infinite speed of signal propagation
-process instabilities
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Causal viscous hydrodynamics
Full tensor decomposition in frame of
W. Israel, J.Stewart 79
The 2nd order Israel-Stewart formulation
--expansion the entropy current to the 2nd order
of dissipative flows,
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Causal viscous hydro in (21)-d
U. Heinz, H. Song A. Chaudhuri, PRC 73(2006)
034904
31 21
-zero net baryon density, zero bulk pressure,
zero heat conduction
Equivalent to the modified transport equations
for
where, the transport equations for energy
momentum tensor are explicit written as
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Viscous vs. ideal hydro temperature entropy
-slowing down of cooling process due to
decelerated longitudinal expansion initially ,
but faster cooling in middle and late stages due
to stronger transverse expansion
-viscous effects are larger in early and middle
stages, but neglectable in late stage
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Viscous vs. ideal hydro radial flow spectra
-More radial flow, flatter spectra
-the viscous effects to the hadron spectra could
be absorbed by starting viscous
hydro later with lower initial energy density
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Viscous vs. ideal hydro elliptical flow
For non-central collisions
-Elliptical flow is very sensitive to even
minimal shear viscosity.
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Summery and Conclusions
- Shear viscosity results in slowing down of the
cooling process in early stage due
to decelerated longitudinal expansion, faster
cooling in middle and late stages due
to stronger transverse expansion.
-Shear viscosity leads to more radial flow
generation and flatter spectra, the effects
of which could be absorbed by starting viscous
hydro later with lower initial energy
density.
-elliptical flow is very sensitive to shear
viscosity, even the minimum value from
AdS/CFT could result in a great suppression of
.

• in contrast with the Teaneys blast wave model
  method, which only counts viscous

spectra corrections during freeze-out
procedure, our results shows that evolution

• corrections ( ) are the main factor to
  influence the low spectra elliptical flow.

-Do the results indicate that the QGP viscosity
smaller than the AdS/CFT lowest
limit? Need more investigations in the at
least flowing aspects
a)Different initial conditions
b) more realistic descriptions of the hadronic
stage
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Thank You
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1st order theory vs. 2nd order theory
As the relaxation time reduces to zero, the 2nd
order theory formulism returns to the 1st order
one. First order theory ,
instable ?
a) calculated with the same velocity
profile from the 2nd theory by

not the real 1st order Navier-Stokes hydrodynamics
b)Smaller relaxation time reduces the viscous
effects on suppression of
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Blast wave model vs. real viscous hydrodynamics
Blast wave model
viscous hydrodynamics
vs.
the1st order theory corrections
(the 2 order theory Israel-Stewart formulation)
--with both evolution and spectra corrections
--with only spectra corrections
Compare the spectra corrections from blast wave
model and from real hydrodynamics
-Spectra corrections are sensitive to the details
of flow pattern at freeze out, and are
not easily captured with blast-wave model
parametrizations.
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viscous vs. ideal hydro central CuCu
collisions
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viscous vs. ideal hydro central CuCu
collisions
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Viscous vs. ideal hydro momentum anisotropy
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Where ideal hydro works

• spectra for both central collision and
  noncentral collisions

Hydro U Heinz P.Kolb Data STAR PHENIX
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Where ideal hydro works

• Elliptic flow coefficient at noncentral
  collisions

STAR, PHENIX, PHOBOS
STAR PRL04 PHENIX PRL03
for different identified hadrons
as a function of centrality
-Ideal hydro describes the data well at
-It also gives a correct mass splitting of
at low region